Question

find the ratio in which the line 3x+y-9=0 divides the line segment joining the points (1,3) and (2,7)

Answer 1

3:4 will be the Ratio

Answer 2

Solution

Given :-

$\sf \implies \: Equation \: 3x + 4y - 9 = 0$

$\sf \implies Point \: (1,3) \: and \: (2,7)$

Let

$\sf \implies \: Ratio \: = P \ratio \: 1$

Using section formula

$\implies \sf \: p \bigg( \dfrac{x_2m + nx_1}{m + n} , \: \dfrac{y_2m + y_1n}{m + n} \bigg)$

Where

$\sf \implies \: x_1 = 1, y_1 = 3 \\ \sf \implies \: x_2 = 2,y_2 = 7$

$\sf \implies \: m = p \: \: and \: n = 1$

Now put the value on formula

$\sf \implies\: p \bigg( \dfrac{2 \times p + 1 \times1}{p + 1} , \: \dfrac{7 \times p + 3 \times 1}{p + 1} \bigg)$

$\sf \implies\: p \bigg( \dfrac{2p + 1 }{p + 1} , \: \dfrac{7 p + 3 }{p + 1} \bigg)$

Now put the value of x and y on Given equation

$\sf \implies \: 3x + 4y - 9 = 0$

$\sf \implies3 \bigg( \dfrac{2p + 1}{p + 1} \bigg) + 4 \bigg( \dfrac{7p + 3}{p + 1} \bigg) - 9 = 0$

$\sf\implies \: \dfrac{6p + 3}{p + 1} + \dfrac{28p + 12}{p + 1} - 9 = 0$

Taking Lcm

$\sf \implies \: \dfrac{6p + 3 + 28p + 12 - 9p - 9}{p + 1} = 0$

$\sf \implies\: 6p + 3 + 28p + 12 - 9p - 9 = 0$

$\sf \implies \: 34p + 15 - 9p - 9 = 0$

$\sf \implies 25p + 6 = 0$

$\sf \implies \: p = \dfrac{ - 6}{25}$

Answer

The ratio is -6/25 or -6:25